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# 1. Definition & Example

• A signed permutation of size $n$ is an bijection $\sigma$ of $\{\pm 1,\ldots,\pm n\}$ such that $\sigma(-i) = -\sigma(i)$.

• We usually denote a signed permutation in one-line notation. This is given by $\pi = [\pi(1),\ldots,\pi(n)]$. E.g., $\pi = [5,-4,2,-3,-1]$ says that

$$\pi(1)=5,\pi(2)=-4,\pi(3)=2,\pi(4)=-3,\pi(5)=-1.$$

 the 8 Signed permutations of size 2 [1,2] [1,-2] [-1,2] [-1,-2] [2,1] [2,-1] [-2,1] [-2,-1]
• There are $2^n\cdot n! = 2^n \cdot 1 \cdot 2 \cdot 3 \cdots n$ signed permutations of size $n$, see A000165.

• The group of signed permutations of size $n$is the Coxeter group of type $B_n$. It is the group of symmetries of a regular hypercube and also known under the name hyperoctahedral group.

TBA

TBA

# 7. Technical information for database usage

• Signed permutations are graded by size.
• The database contains all signed permutations of size at most 5.